Theorem breq12 4064

Description: Equality theorem for a binary relation. (Contributed by NM, 8-Feb-1996.)

Assertion

Ref	Expression
breq12	|- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Proof of Theorem breq12

Step	Hyp	Ref	Expression
1		breq1 4062	. 2 |- ( A = B -> ( A R C <-> B R C ) )
2		breq2 4063	. 2 |- ( C = D -> ( B R C <-> B R D ) )
3	1, 2	sylan9bb 462	1 |- ( ( A = B /\ C = D ) -> ( A R C <-> B R D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   class class class wbr 4059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060

This theorem is referenced by:  breq12i  4068  breq12d  4072  breqan12d  4075  posng  4765  isopolem  5914  poxp  6341  rbropapd  6351  ecopover  6743  ecopoverg  6746  ltdcnq  7545  recexpr  7786  ltresr  7987  reapval  8684  ltxr  9932  xrltnr  9936  xrltnsym  9950  xrlttr  9952  xrltso  9953  xrlttri3  9954  xposdif  10039  f1olecpbl  13260  exmidsbthrlem  16163